An aircraft manufacturing company wants to optimize their products for passenger airlines. The company's latest research shows that most of the delays happen because of slow boarding.

Most of the medium-sized aircraft are designed with 3-3 seat layout, meaning each row has 6 seats: 3 seats on the left side, a single aisle, and 3 seats on the right side. At each of the left and right sides there is a window seat, a middle seat, and an aisle seat. A passenger that boards an aircraft assigned to an aisle seat takes significantly less time than a passenger assigned to a window seat even when there is no one else in the aircraft.

The company decided to compute an inconvenience of a layout as the total sum of distances from each of the seats of a single row to the closest aisle. The distance from a seat to an aisle is the number of seats between them. For a 3-3 layout, a window seat has a distance of 2, a middle seat — 1, and an aisle seat — 0. The inconvenience of a 3-3 layout is $$$(2+1+0)+(0+1+2)=6$$$. The inconvenience of a 3-5-3 layout is $$$(2+1+0)+(0+1+2+1+0)+(0+1+2)=10$$$.

Formally, a layout is a sequence of positive integers $$$a_1, a_2, \ldots, a_{k+1}$$$ — group $$$i$$$ having $$$a_i$$$ seats, with $$$k$$$ aisles between groups, the $$$i$$$-th aisle being between groups $$$i$$$ and $$$i+1$$$. This means that in a layout each aisle must always be between two seats, so no aisle can be next to a window, and no two aisles can be next to each other.

The company decided to design a layout with a row of $$$n$$$ seats, $$$k$$$ aisles and having the minimum inconvenience possible. Help them find the minimum inconvenience among all layouts of $$$n$$$ seats and $$$k$$$ aisles, and count the number of such layouts modulo $$$998\,244\,353$$$.